Micro- and nanorobots are often controlled by global input signals, such as an
electromagnetic or gravitational field. These fields move each robot maximally
until it hits a stationary obstacle or another stationary robot. This paper
investigates 2D motion-planning complexity for large swarms of simple mobile
robots (such as bacteria, sensors, or smart building material).

In previous work we proved it is NP-hard to decide whether a given initial
configuration can be transformed into a desired target configuration; in this
paper we prove a stronger result: the problem of finding an optimal control
sequence is PSPACE-complete. On the positive side, we show we can build useful
systems by designing obstacles. We present a reconfigurable hardware platform
and demonstrate how to form arbitrary permutations and build a compact
absolute encoder. We then take the same platform and use dual-rail
logic to build a universal logic gate that concurrently evaluates AND,
NAND, NOR and OR operations. Using many of these gates and appropriate
interconnects we can evaluate any logical expression.